| L(s) = 1 | + 2·2-s + 3·3-s + 3·4-s + 6·6-s − 3·7-s + 4·8-s + 6·9-s + 3·11-s + 9·12-s − 5·13-s − 6·14-s + 3·16-s + 4·17-s + 12·18-s − 19-s − 9·21-s + 6·22-s + 12·24-s − 10·26-s + 10·27-s − 9·28-s + 2·29-s + 17·31-s + 6·32-s + 9·33-s + 8·34-s + 18·36-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.73·3-s + 3/2·4-s + 2.44·6-s − 1.13·7-s + 1.41·8-s + 2·9-s + 0.904·11-s + 2.59·12-s − 1.38·13-s − 1.60·14-s + 3/4·16-s + 0.970·17-s + 2.82·18-s − 0.229·19-s − 1.96·21-s + 1.27·22-s + 2.44·24-s − 1.96·26-s + 1.92·27-s − 1.70·28-s + 0.371·29-s + 3.05·31-s + 1.06·32-s + 1.56·33-s + 1.37·34-s + 3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.61591510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.61591510\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $D_{6}$ | \( 1 - p T + T^{2} + p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 3 T + 2 p T^{2} + 25 T^{3} + 2 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 5 T + 3 p T^{2} + 122 T^{3} + 3 p^{2} T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 26 T^{2} - 158 T^{3} + 26 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + T + 2 p T^{2} + 63 T^{3} + 2 p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 26 T^{2} + 58 T^{3} + 26 p T^{4} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 63 T^{2} - 132 T^{3} + 63 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 17 T + 181 T^{2} - 1190 T^{3} + 181 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 36 T^{2} - 34 T^{3} + 36 p T^{4} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 4 T + 122 T^{2} - 326 T^{3} + 122 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 17 T + 97 T^{2} + 362 T^{3} + 97 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 10 T + 166 T^{2} - 948 T^{3} + 166 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 11 T^{2} + 188 T^{3} + 11 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 6 T + 146 T^{2} + 572 T^{3} + 146 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 3 T + 95 T^{2} + 10 p T^{3} + 95 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 7 T + 9 T^{2} - 650 T^{3} + 9 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 26 T + 430 T^{2} - 4272 T^{3} + 430 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 10 T + 175 T^{2} - 988 T^{3} + 175 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 6 T + 200 T^{2} - 736 T^{3} + 200 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 6 T + 33 T^{2} - 4 p T^{3} + 33 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2 T + 167 T^{2} - 28 T^{3} + 167 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 29 T + 366 T^{2} + 3473 T^{3} + 366 p T^{4} + 29 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.348649416063764158760393437178, −8.758315747985887597800622329874, −8.430644717312248673885648608673, −8.223247391396777714175356612397, −7.86609369650864828657033254658, −7.79553841138608589125578986841, −7.28489429468389571638328499240, −6.95060357752092697923653229552, −6.63666534237307394030041054888, −6.50659426085273659986767504246, −6.43591438537674469624360095397, −5.80716878558180185936144345946, −5.48954106279152338801918672356, −4.87809759700423328069375888932, −4.81317744658448132552956067139, −4.49763813187114139855412570437, −4.05657716670590664058501119166, −3.77371419785787475240012480777, −3.39406231718069596091459456266, −3.04866917193860400925149613291, −2.80874805352359110110031403542, −2.43667557594614938349342832539, −2.14848788018557771862822690356, −1.45428727031405064443889610082, −0.853024506382851953622121402073,
0.853024506382851953622121402073, 1.45428727031405064443889610082, 2.14848788018557771862822690356, 2.43667557594614938349342832539, 2.80874805352359110110031403542, 3.04866917193860400925149613291, 3.39406231718069596091459456266, 3.77371419785787475240012480777, 4.05657716670590664058501119166, 4.49763813187114139855412570437, 4.81317744658448132552956067139, 4.87809759700423328069375888932, 5.48954106279152338801918672356, 5.80716878558180185936144345946, 6.43591438537674469624360095397, 6.50659426085273659986767504246, 6.63666534237307394030041054888, 6.95060357752092697923653229552, 7.28489429468389571638328499240, 7.79553841138608589125578986841, 7.86609369650864828657033254658, 8.223247391396777714175356612397, 8.430644717312248673885648608673, 8.758315747985887597800622329874, 9.348649416063764158760393437178
